Properties

Label 2-6864-1.1-c1-0-3
Degree $2$
Conductor $6864$
Sign $1$
Analytic cond. $54.8093$
Root an. cond. $7.40333$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 2.39·5-s − 1.05·7-s + 9-s − 11-s + 13-s + 2.39·15-s − 5.96·17-s + 5.05·19-s + 1.05·21-s − 9.15·23-s + 0.740·25-s − 27-s − 2.65·29-s + 9.70·31-s + 33-s + 2.51·35-s − 4.51·37-s − 39-s + 3.57·41-s − 7.96·43-s − 2.39·45-s + 1.20·47-s − 5.89·49-s + 5.96·51-s − 0.689·53-s + 2.39·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.07·5-s − 0.397·7-s + 0.333·9-s − 0.301·11-s + 0.277·13-s + 0.618·15-s − 1.44·17-s + 1.15·19-s + 0.229·21-s − 1.90·23-s + 0.148·25-s − 0.192·27-s − 0.493·29-s + 1.74·31-s + 0.174·33-s + 0.425·35-s − 0.742·37-s − 0.160·39-s + 0.557·41-s − 1.21·43-s − 0.357·45-s + 0.176·47-s − 0.842·49-s + 0.835·51-s − 0.0946·53-s + 0.323·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(6864\)    =    \(2^{4} \cdot 3 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(54.8093\)
Root analytic conductor: \(7.40333\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{6864} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 6864,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5204108951\)
\(L(\frac12)\) \(\approx\) \(0.5204108951\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
11 \( 1 + T \)
13 \( 1 - T \)
good5 \( 1 + 2.39T + 5T^{2} \)
7 \( 1 + 1.05T + 7T^{2} \)
17 \( 1 + 5.96T + 17T^{2} \)
19 \( 1 - 5.05T + 19T^{2} \)
23 \( 1 + 9.15T + 23T^{2} \)
29 \( 1 + 2.65T + 29T^{2} \)
31 \( 1 - 9.70T + 31T^{2} \)
37 \( 1 + 4.51T + 37T^{2} \)
41 \( 1 - 3.57T + 41T^{2} \)
43 \( 1 + 7.96T + 43T^{2} \)
47 \( 1 - 1.20T + 47T^{2} \)
53 \( 1 + 0.689T + 53T^{2} \)
59 \( 1 + 11.4T + 59T^{2} \)
61 \( 1 + 6.89T + 61T^{2} \)
67 \( 1 - 14.2T + 67T^{2} \)
71 \( 1 - 8.10T + 71T^{2} \)
73 \( 1 - 2.25T + 73T^{2} \)
79 \( 1 - 11.4T + 79T^{2} \)
83 \( 1 + 15.4T + 83T^{2} \)
89 \( 1 + 11.8T + 89T^{2} \)
97 \( 1 + 14.1T + 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.111383522626128079900481980170, −7.20628813557395946045709992809, −6.57272088860453203776393299414, −5.95286009348591616409175772402, −5.05949708288702806537251648419, −4.31999920985390292643428866573, −3.74203375663625967250108933134, −2.84564189069411439058468649375, −1.71913193089937421582851178747, −0.37186469345270579816772238051, 0.37186469345270579816772238051, 1.71913193089937421582851178747, 2.84564189069411439058468649375, 3.74203375663625967250108933134, 4.31999920985390292643428866573, 5.05949708288702806537251648419, 5.95286009348591616409175772402, 6.57272088860453203776393299414, 7.20628813557395946045709992809, 8.111383522626128079900481980170

Graph of the $Z$-function along the critical line